Graphics Reference
In-Depth Information
(
)(
)
=
()(
)
+
()( )
Pp u v
,
b u p
0
,
v
b u p
1
,
v
,
(12.21a)
1
0
1
(
)(
)
=
()(
)
+
()(
)
Pp uv
,
c vpu
,
0
c vpu
,
1
.
(12.21b)
2
0
1
Notice how the definition of the functions (P
i
p)(u,v) only uses the boundary values of
p(u,v). These new equations reduce to the ones in equation (12.19) if we let b
0
(t) =
c
0
(t) = L
0,1
(t) and b
1
(t) = c
1
(t) = L
1,1
(t), where L
0,1
(t) = 1 - t and L
1,1
(t) = t are the linear
Lagrange basis functions defined in equation (11.2) of Section 11.2.1. In fact, the only
conditions that the blending functions b
i
(t) and c
i
(t) have to satisfy in order to have
the functions defined by equation (12.21) interpolate the v-, respectively, u-direction
boundary are that
()
=
()
=
()
=
()
=
b
01
,
b
10
,
b
0 0
,
b
11
,
0
0
1
1
()
=
()
=
()
=
()
=
c
01
,
c
10
,
c
0 0
,
c
11
,
0
0
1
1
The operators P
1
and P
2
are best thought of as projection operators that project
the space of vector-valued functions of two variables onto a subspace because P
i
2
=
P
i
. The sum of the two operators is not a projection since (P
1
+ P
2
)
2
π P
1
+ P
2
, but if
they commute, that is, P
1
P
2
p = P
2
P
1
p, then the
Boolean sum operator
P defined by
(
)
(
)
Pp
=≈
P
P
p
=+-
P
P
P P
p
(12.22a)
1
2
1
2
1
2
is
a projection. Equation (12.22a) can be expressed in matrix form as
-
( )
-
() ( )
-
()
-
()
()
()
p
00
,
p
01
,
p
0
,
v
cv
cv
Ê
ˆ
Ê
ˆ
0
Á
Á
˜
˜
Á
Á
˜
˜
( ( )
=
(
() ()
)
( )
Pp u v
,
b
u b
u
1
p
10
,
p
11
,
p
1
,
v
(12.22b)
0
1
1
Ë
¯
Ë
¯
(
)
(
)
pu
,
0
pu
,
1
0
1
.
We summarize the main facts about the operator P.
12.7.1
Theorem.
If the functions b
i
(u) and c
i
(v) and operators P
i
in equations
(12.21) satisfy
(1) b
i
(j) = c
i
(j) =d
ij
, i,j Π{0,1}, and
(2) P
1
P
2
p = P
2
P
1
p for all p,
then the parametric surface (Pp)(u,v) defined by equations (12.22) interpolates the
boundary curves p(0,v), p(1,v), p(u,0), and p(u,1). If also
(3) b
i
¢(j) = c
i
¢(j) = 0, i,j Œ {0,1},
then
∂
∂
∂
∂
p
u
∂
∂
p
u
( ( )
=
()
( )
+
()
( )
Pp u v
,
c
v
i
,
0
c
v
i
,
1
0
1
u
∂
∂
∂
∂
p
v
∂
∂
p
v
( ( )
=
()
( )
+
()
( )
Pp u v
,
b
v
i
,
0
bv
i
,
1
,
0
1
v
